Systems and methods for printing of 3d models

ABSTRACT

There is provided a method of representing a three dimensional (3D) object using univariate curves, comprising: receiving an initial definition of a 3D object representation, calculate a covering set of univariate curves, the covering set comprising at least one non-planar univariate curve, wherein the covering set of univariate curves represent the volume of the 3D object within a tolerance requirement, and generating a representation of the 3D object based on the set of univariate curves, wherein the set of univariate curves represent the volume of the 3D object.

BACKGROUND

The present invention, in some embodiments thereof, relates to geometricmodeling and, more specifically, but not exclusively, to 3D printing ofgeometric models.

Additive manufacturing (AM) methods are based on dividing a 3D model(often represented as a polygonal mesh) into a series of two dimensionalparallel planes (parallel to the printing surface of the 3D printer),also termed slices. The 3D printer deposits material according to eachslice, printing the 3D model from the bottom up, one layer at a time.Additional detail of slicing based AM systems may be found, for example,with reference to Pulak Mohan Pandey, N Venkata Reddy, and Sanjay GDhande. Slicing procedures in layered manufacturing: a review. Rapidprototyping journal, 9(5):274-288, 2003.

For example, the surface finish and/or the strength of the object is notoptimal, being influenced by the slicing orientation and/or theprint-path used to print the slice.

Better 3D printing systems and methods are being developed to improvethe quality of the 3D printed object.

According to an aspect of some embodiments of the present inventionthere is provided a method of representing the volume of a threedimensional (3D) object, comprising: receiving an initial definition ofa 3D object representation; calculating a covering set of univariatecurves, the covering set comprising at least one non-planar univariatecurve, wherein the covering set of univariate curves represent thevolume of the 3D object within a tolerance requirement; and generating arepresentation of the 3D object based on the set of univariate curves,wherein the set of univariate curves represent the volume of the 3Dobject.

Optionally, the covering set of univariate curves are computed based ona plurality of non-planar bivariate parametric surfaces covering whereinthe 3D object, wherein the 3D object comprises at least one trivariate,within the tolerance requirement, and further comprising fitting each ofthe plurality of surfaces with at least one univariate curve.

Optionally, the method further comprises defining at least one externaldirectional vector field enclosing the 3D object, and wherein coveringcomprises fitting each of the plurality of curves according to the atleast one external directional vector field to create the covering setof univariate curves. Optionally, the method further comprises clippingthe plurality of curves to be inside the boundaries of the 3D object.

Optionally, covering comprises dividing the volume enclosed by theboundaries of the 3D representation into a plurality of discretevolumetric cells, covering each cell by tracing a freeform curve fromthe center of each respective discrete cell, wherein all accessiblecells without any other cell below them are uniquely covered by curvesand removed from the 3D object, until no more uncovered cells remain,and further comprising terminating the tracing of the freeform curvewhen another covered discrete cell is encountered or when the freeformcurve leaves the volume of the respective discrete cell. Optionally, thevolume enclosed by the boundaries of the 3D representation is dividedinto dexel (depth) cells. Optionally, the freeform curve is tracedfollowing a direction according to an external vector field.

Optionally, covering comprises creating a set of univariate curves inthe domain of each trivariate in the 3D object and mapping the curves,using curve-trivariate function composition, to Euclidean space.

Optionally, the 3D object comprises at least one trimmed trivariate.

Optionally, the method further comprises printing the 3D object by anadditive manufacturing system using the set of univariate curves asprinting paths.

Optionally, the method further comprises performing at least one offiber placement and composite material manufacturing using the set ofunivariate curves as layout paths.

Optionally, the plurality of non-planar bivariate parametric surfacesare iso-parametric surfaces of a trivariate parametric volumerepresentation of the 3D object representation.

Optionally, the at least one univariate curve is an iso-parametric curveof a trivariate parametric volume representation of the 3D objectrepresentation.

Optionally, at least one additional external field represents one ormore property selected from the group consisting of: color,transparency, a surface finish, a mechanical strength or stiffness, andgraded-, anisotropic-, heterogeneous-material properties, and each ofsuch field further comprising assigning to each curve-member of the setof covering univariate curves, a value for a parameter denoting thefield's value when the respective curve-member is used as a printingpath, wherein the parameter is variable along the length of eachcurve-member.

Optionally, the method further comprises ordering the set of univariatecurves for printing by a 3D printing head along each curve-member of theset of univariate curves, wherein the ordering is selected such thatduring the 3D printing process curves that are not yet printed and haveno unprinted curves below them are printed.

Optionally, the method further comprises computing a directed graph thatstores the ordering of printing the set of univariate curves, whereineach curve-member is assigned to a vertex of the directed graph, and adirected edge between two vertexes exists when a first of the twovertexes is below a second of the two vertexes relative to the 3Dprinting head, in full 3-axis additive manufacturing 3D printing,wherein when the directed graph is acyclic the directed graph representsa 3D printable ordering of the set of univariate curves.

Alternatively or additionally, the method further comprises computing adirected graph that stores the ordering of printing the set ofunivariate curves, wherein each curve-member is assigned to a vertex ofthe directed graph, and a directed edge between two vertexes exists whena first of the two vertexes is below a second of the two vertexesrelative to the 3D printing head, in 5-axis additive manufacturing 3Dprinting, wherein when the directed graph is acyclic the directed graphrepresents a 3D printable ordering of the set of univariate curves.

Alternatively or additionally, the method further comprises identifyinga cycle in the directed graph that represents an impossible-to-printsubset of univariate curves, and dividing at least one of the subset ofunivariate curves to remove the cycle from the directed graph.

Optionally, the directed graph is computed by dividing each curve-memberof the set of univariate curves into a plurality of fragments eachhaving a length less than a fragmentation tolerance, and iterativelymerging pairs of the plurality of fragments while maintaining an acyclicdirected graph. Optionally, the fragmentation tolerance corresponds to aminimum height of material deposited by the 3D printing head.

Optionally, the method further comprises assigning to each curve-memberof the set of univariate curves, a value for a parameter denoting aneffective coverage radius indicative of the amount of local material fordeposition by a 3D printing head when the respective curve-member isused as a printing path, wherein the parameter is variable along thelength of each curve-member.

Optionally, the method further comprises defining a support volume forthe set of covering univariate curves, and representing the supportvolume using support elements, wherein the support elements are printedbefore printing a portion of the set of univariate curves.

According to an aspect of some embodiments of the present inventionthere is provided a system for processing of a three dimensional (3D)object for manufacturing, comprising: a program store storing code; anda processor coupled to the program store for implementing the storedcode, the code comprising: code to receive an initial definition of the3D object, calculate a covering set of univariate curves, the coveringset comprising at least one non-planar univariate curve, wherein thecovering set of univariate curves represent the volume of the 3D objectwithin a tolerance requirement; and generate a representation of the 3Dobject based on the covering set of univariate curves, wherein thecovering set of univariate curves represents the volume defined by the3D object.

According to an aspect of some embodiments of the present inventionthere is provided a computer program product comprising a non-transitorycomputer readable storage medium storing program code thereon forimplementation by a processor of a system for processing of a 3D object,comprising: instructions to receive an initial definition of the 3Dobject; instructions to calculate a covering set of univariate curves,the covering set comprising at least one non-planar univariate curve,wherein the covering set of univariate curves represent the volume ofthe 3D object within a tolerance requirement; and instructions togenerate a representation of the 3D object based on the covering set ofunivariate curves, wherein the covering set of univariate curvesrepresent the volume define by the 3D object.

Unless otherwise defined, all technical and/or scientific terms usedherein have the same meaning as commonly understood by one of ordinaryskill in the art to which the invention pertains. Although methods andmaterials similar or equivalent to those described herein can be used inthe practice or testing of embodiments of the invention, exemplarymethods and/or materials are described below. In case of conflict, thepatent specification, including definitions, will control. In addition,the materials, methods, and examples are illustrative only and are notintended to be necessarily limiting.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

Some embodiments of the invention are herein described, by way ofexample only, with reference to the accompanying drawings. With specificreference now to the drawings in detail, it is stressed that theparticulars shown are by way of example and for purposes of illustrativediscussion of embodiments of the invention. In this regard, thedescription taken with the drawings makes apparent to those skilled inthe art how embodiments of the invention may be practiced.

In the drawings:

FIG. 1A is a flowchart of a method for generating a set of univariatecurves that fill (i.e., cover) the volume of a 3D geometric object, inaccordance with some embodiments of the present invention;

FIG. 1B is a flowchart of a method for generating the set of univariatecurves, providing additional details to the method described withreference to FIG. 1A, in accordance with some embodiments of the presentinvention;

FIG. 2 is a block diagram of a system that generates the set ofunivariate curves for 3D printing of the 3D geometric object, inaccordance with some embodiments of the present invention;

FIG. 3 is pseudo code of a method for covering a trivariate volume usingiso-parametric surfaces of the volume, in accordance with someembodiments of the present invention;

FIG. 4 is a schematic depicting the process of starting with a volume,covering the volume with surfaces, and fitting curves to the surfaces,in accordance with some embodiments of the present invention;

FIG. 5 is a schematic that depicts a coverage of a trivariate volumecovered by surfaces with different values of the tolerance requirement,in accordance with some embodiments of the present invention;

FIG. 6 is a schematic that depicts an example of a situation in whichtwo surfaces are close together, but the volume in between the surfacesincludes points that are too far from both surfaces, in accordance withsome embodiments of the present invention;

FIG. 7 is a schematic where a new distance measure may work better thanan overly conservative measure, in accordance with some embodiments ofthe present invention;

FIG. 8 is a schematic depicting an example of a helical 3D object and aset of univariate curves created using the systems and/or methodsdescribed herein, in accordance with some embodiments of the presentinvention;

FIG. 9 is a schematic of an exemplary trimmed helical trivariate volumethat is created by intersecting the trivariate helix of FIG. 8 with atrivariate sphere, in accordance with some embodiments of the presentinvention;

FIG. 10 is a schematic depicting the process of clipping the set ofcurves according to an external vector field, in accordance with someembodiments of the present invention;

FIG. 11 includes schematics depicting the process of mapping curves intoEuclidean space, in accordance with some embodiments of the presentinvention;

FIG. 12 is a schematic of the process of creating the set of curves thatare used to 3D print the geometric object, in accordance with someembodiments of the present invention;

FIG. 13 is a schematic representing the geometry of the printing headused to compute the accessibility and the order of the set of curves, inaccordance with some embodiments of the present invention;

FIG. 14 is a schematic depicting an intersection between a volumedefined by a sweep of a cone bounding the printing head during printingof a lower curve, and a portion of an upper curve, in accordance withsome embodiments of the present invention;

FIG. 15 is a schematic that graphically depicts the process of mergingsub-curves (and/or fragments), in accordance with some embodiments ofthe present invention;

FIG. 16 is a schematic depicting the process of ordering curves forprinting a helical 3D object covered by a set of nine curves, inaccordance with some embodiments of the present invention;

FIG. 17 is a schematic depicting the process of selecting the material'sdeposition radius for curves, in accordance with some embodiments of thepresent invention;

FIG. 18 is a schematic depicting support elements, in accordance withsome embodiments of the present invention; and

FIG. 19 includes schematics depicting the differences between thecomputed covering set of curves for a 3D object represented using atrivariate and the covering set of curves computed using the externalvector field and clipping for a general 3D object, in accordance withsome embodiments of the present invention.

DETAILED DESCRIPTION

The present invention, in some embodiments thereof, relates to geometricmodeling and, more specifically, but not exclusively, to covering(filling) the volume of a 3D geometric object using curves, optionallytowards additive manufacturing (AM) of geometric models.

As used herein, the terms cover and fill are sometimes interchanged.

An aspect of some embodiments of the present invention relate to systemsand/or methods (e.g., code instructions stored in a storage deviceexecuted by processor(s)) that generate a representation of a 3Dgeometric object using a set of univariate curves that fill and/or cover(within a tolerance requirement) the volume of the 3D geometric object.The univariate curves are general 3-space curves, do not necessarily liein parallel planes, and/or the univariate curves are not necessarilyparallel to each other. The univariate curves may define, for example,printing paths for a printing head of a three dimensional (3D) printerin an additive manufacturing (AM) machine.

Optionally, an initial definition of the 3D geometric object, optionallya (possibly trimmed) trivariate parametric volume representation, iscovered (within the tolerance requirement) with non-planar and/ornon-parallel bivariate parametric surfaces. Each of the surfaces isfitted with one or more univariate curves to create the covering set ofunivariate curves to the surface.

Optionally, the covering univariate curves are fitted to the 3Dgeometric object according to an external directional vector field(sometimes referred to herein as a directional field) that encloses the3D object. The external vector field may define the directions of thecurves. The curves are clipped according to the 3D object, such that theremaining curves are located within the 3D object

Alternatively or additionally, the univariate curves are specified inthe domain of the 3D geometric object, represented as a set of(optionally trimmed) trivariates, and a curve-trivariate functioncomposition that is mapping the univariate covering set into Euclideanspace.

Alternatively or additionally, the 3D volume enclosing the 3D object isdivided into discrete volumetric cells, optionally as Dexels. Free formcurve(s) are traced from the center (or another location) within eachrespective discrete cell.

Optionally, additional fields can specify values inside the 3D object,for example, color, transparency, material properties, and/or mechanicalstiffness requirements.

Optionally, the curve-members of the set of univariate curves areordered for printing by the 3D printing head. The ordering is based onprinting the lower curves before higher curves. The ordering may bedefined by a directed acyclic graph, where each vertex represents one ormore curves and the directed edges represent a printing order betweenthe curves of the respective vertexes. Cycles in the directed graph,which represent an impossible-to-print subset of univariate curves(e.g., printing of higher level curves before lower level curves, whichis impossible without the printer head penetrating and damaging thehigher level curve to print the lower level curve) may be removed bydividing at least one of the univariate curves in the subset.

The systems and/or methods (e.g., code instructions stored in a storagedevice executed by processor(s)) provided herein provide, as an example,a technical solution to the technical problem of improving the processof 3D printing (i.e., AM) a 3D geometric object, optionally a trivariateparametric volume representation. The technical problem may includeimprovement of the surface finish, mechanical strength, and/or materialproperties of the 3D printed object.

The systems and/or methods (e.g., code instructions stored in a storagedevice executed by processor(s)) provided herein improve performance ofa 3D printing system (e.g., AM system or other systems that manufacturea 3D object) by creating a set of univariate curves based on the 3Dgeometric object representation, that when printed improve the surfacefinish, mechanical strength, and/or material properties of the 3Dprinted object. The improvement is in comparison to, for example,standard 3D printing systems that divide the 3D geometric objectrepresentation into multiple parallel slices, and print one slice at atime, from the bottom up.

Accordingly, the systems and/or methods described herein areinextricably tied to computer technology and 3D printing (i.e., AM)technology to overcome an actual technical problem arising in 3Dprinting of geometric objects.

Before explaining at least one embodiment of the invention in detail, itis to be understood that the invention is not necessarily limited in itsapplication to the details of construction and the arrangement of thecomponents and/or methods set forth in the following description and/orillustrated in the drawings and/or the Examples. The invention iscapable of other embodiments or of being practiced or carried out invarious ways.

The present invention may be a system, a method, and/or a computerprogram product. The computer program product may include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, and any suitable combination of theforegoing. A computer readable storage medium, as used herein, is not tobe construed as being transitory signals per se, such as radio waves orother freely propagating electromagnetic waves, electromagnetic wavespropagating through a waveguide or other transmission media (e.g., lightpulses passing through a fiber-optic cable), or electrical signalstransmitted through a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, or either source code or object code written in anycombination of one or more programming languages, including an objectoriented programming language such as Smalltalk, C++ or the like, andconventional procedural programming languages, such as the “C”programming language or similar programming languages. The computerreadable program instructions may execute entirely on the user'scomputer, partly on the user's computer, as a stand-alone softwarepackage, partly on the user's computer and partly on a remote computeror entirely on the remote computer or server. In the latter scenario,the remote computer may be connected to the user's computer through anytype of network, including a local area network (LAN) or a wide areanetwork (WAN), or the connection may be made to an external computer(for example, through the Internet using an Internet Service Provider).In some embodiments, electronic circuitry including, for example,programmable logic circuitry, field-programmable gate arrays (FPGA), orprogrammable logic arrays (PLA) may execute the computer readableprogram instructions by utilizing state information of the computerreadable program instructions to personalize the electronic circuitry,in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the block may occur out of theorder noted in the figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

As used herein, the term additive manufacturing, and 3D printing areinterchangeable.

As used herein, the term object and geometric object areinterchangeable.

Inventors realized that traditional AM manufacturing methods that slicethe representation of the 3D object into parallel planes create slicesthat are not intrinsic to the original 3D object representation, and mayresult in inferior mechanical strength, material properties, surfacesfinishes, and the like, that would otherwise be possible.

The systems and/or methods (e.g., code executable by processor(s))described herein that create the set of univariate curves, addresstechnical challenges that are not solved by other methods. Althoughother attempts have been made to print 3D objects without slicing, suchmethods do not provide complete coverage of the volume of the 3D objectand leave unsolved technical challenges that are addressed by thesystems and/or methods described herein. The exemplary other attemptsare based on dictating printing directions that are independent of themodel, or the methods are limited to specific simple geometries.

For example, Stefanie Mueller, Sangha Im, Serafima Gurevich, AlexanderTeibrich, Lisa Pfisterer, Francois Guimbretiere, and Patrick Baudisch.Wireprint: 3d printed previews for fast prototyping. In Proceedings ofthe 27th annual ACM symposium on User interface software and technology,pages 273-280. ACM, 2014, appears to describe how a wireframe on theboundary of an object can be printed directly without slicing to quicklycreate a visual representation of the object. However, only a wireframerepresentation of the boundary is created by this method.

For example, Dave Bourell, Brent Stucker, Yong Chen, Chi Zhou, andJingyuan Lao. A layerless additive manufacturing process based on cncaccumulation. Rapid Prototyping Journal, 17(3):218-227, 2011, and YayuePan, Chi Zhou, Yong Chen, and Jouni Partanen. Multitool and multi-axiscomputer numerically controlled accumulation for fabricating conformalfeatures on curved surfaces. Journal of Manufacturing Science andEngineering, 136(3):031007, 2014, appear to described how manuallyplanned print-paths are used to create objects without slicing. Themethod relies on manual definitions by the user, and the focus appearsto be on implementing and testing the hardware needed to accomplish theprints.

For example, Bin Huang, Sarat B Singamneni, R Ian Campbell, and IanGibson. Curved layer adaptive slicing (clas) for fused depositionmodelling. Rapid Prototyping Journal, 21(4), 2015, appears to relate tonon-planar slices that are created and used to print special geometries.However, not all objects are printable. The objects are limited togeometries that can be expressed as offsets of polygonal faces (e.g., asdescribed with reference to Xiuzhi Qu and Brent Stucker. A 3d surfaceoffset method for stl-format models. Rapid Prototyping Journal,9(3):133-141, 2003), starting from the top (facing up) polygonal surfaceof the object. For example a sphere cannot be printed since the bottomis not an offset of the top, while an arch shaped bridge structure canbe printed. The mentioned future work seeks to handle more complexgeometries by subdividing an object, however, the division is into twogroups: parts that can be printed using non-planar slices using thedescribed method, and complex parts that are still printed usingtraditional slicing methods.

For example, Wei Gao, Yunbo Zhang, Diogo C Nazzetta, Karthik Ramani, andRaymond J Cipra. Revomaker: Enabling multi-directional andfunctionally-embedded 3d printing using a rotational cuboidal platform.In Proceedings of the 28th Annual ACM Symposium on User InterfaceSoftware & Technology, pages 437-446. ACM, 2015, appears to introduce anadditional degree of freedom to 3D printing, and allow printing around acuboid (box) object and creating slices in six different directionscorresponding to the facets of the cuboid. However, the resulting objectis, in essence, made of six traditionally sliced (and manufactured)objects fused together.

Reference is now made to FIG. 1A, which is a flowchart of a method forgenerating a representation of a 3D geometric object as a set ofunivariate curves that fill the volume of the 3D geometric objectaccording to a tolerance requirement, in accordance with someembodiments of the present invention. FIG. 1B discussed below providesadditional features to the method described with reference to FIG. 1A.Reference is also made to FIG. 2, which is a block diagram of a system200 that generates the set of univariate curves for 3D printing of the3D geometric object using the univariate curves as print paths for aprinting head of a 3D printer 202, in accordance with some embodimentsof the present invention. System 200 may execute the acts of the methoddescribed with reference to FIG. 1A, for example, by a processing unit204 of a computing unit 206 executing code instructions stored in aprogram store 208.

The systems and/or methods described herein are designed to be generallyapplicable to any closed 3D object, by offering an intrinsic set of 3Dprinting paths based on the fitted univariate curves. The systems and/ormethods described herein may process a general model provided as input,by automatically filling the entire volume (within the tolerancerequirement) of the model with curves that fit each model's intrinsicspecific structure and intrinsic requirement.

3D printer 202 prints 3D objects by depositing materials using a printerhead that deposits materials. 3D printer 202 may be a standard existingprinter (for example, designed to print parallel slices to create the 3Dobject, one slice at a time, from lower layers to higher layers) that isprogrammed with improved printing paths based on the generated set ofunivariate curves, as described herein.

Computing unit 206 may be integrated with 3D printer 202, for example,as a control console and/or control unit and/or instructions code storedwithin 3D printer 202. Alternatively or additionally, computer unit 206may be implemented as a unit external to 3D printer 202, that createsthe set of univariate curves that are provided (e.g., transmitted usinga communication channel such as a network, or obtained from a storagedevice) to 3D printer 202 for use in guiding the printing of the 3Dobject. For example, computing unit 206 may be implemented as, forexample, a client terminal, a server, a mobile device, a desktopcomputer, a thin client, a Smartphone, a Tablet computer, a laptopcomputer, a wearable computer, glasses computer, and a watch computer.Computing unit 206 may include locally stored software that performedone or more of the acts described with reference to FIG. 1A-B, and/ormay act as one or more servers (e.g., network server, web server, acomputing cloud) that provides services (e.g., one or more of the actsdescribed with reference to FIG. 1A-B) to one or more client terminals,for example, providing software as a service (SaaS) to the clientterminal(s), providing an application for local download to the clientterminal(s), and/or providing functions using a remote access session tothe client terminals, such as through a web browser.

Processing unit 204 may be implemented, for example, as a centralprocessing unit(s) (CPU), a graphics processing unit(s) (GPU), fieldprogrammable gate array(s) (FPGA), digital signal processor(s) (DSP),and application specific integrated circuit(s) (ASIC). Processingunit(s) 204 may include one or more processors (homogenous orheterogeneous), which may be arranged for parallel processing, asclusters and/or as one or more multi core processing units.

Program store 208 stores code instructions implementable by processingunit 204, for example, a random access memory (RAM), read-only memory(ROM), and/or a storage device, for example, non-volatile memory,magnetic media, semiconductor memory devices, hard drive, removablestorage, and optical media (e.g., DVD, CD-ROM). For example, programstore 208 may store code instructions 208A that executes one or moreacts of the method described with reference to FIG. 1A-B.

Computing unit 206 may include a data repository 210 for storing data,for example, a computer aided design (CAD) application for use by a userto design the geometric object for printing, heuristic analysis codeused to improve the computational efficiency of generating the set ofunivariate curves. Data repository 210 may be implemented as, forexample, a memory, a local hard-drive, a removable storage unit, anoptical disk, a storage device, and/or as a remote server and/orcomputing cloud (e.g., accessed using a network connection). It is notedthat code instructions 208A may be stored in data repository 210, forexample, with executing portions loaded into program store 208 forexecution by processing unit 204.

Computing unit 206 may include a network interface 212 for connecting toa network 214, for example, one or more of, a network interface card, awireless interface to connect to a wireless network, a physicalinterface for connecting to a cable for network connectivity, a virtualinterface implemented in software, network communication softwareproviding higher layers of network connectivity, and/or otherimplementations. Computing unit 206 may access one or more remoteservers 216 using network 214, for example, to download the initialdefinition of the geometric object, and/or to provide the generated setof univariate curves.

Computing unit 206 may connect using network 214 (or anothercommunication channel, such as through a direct link (e.g., cable,wireless) and/or indirect link (e.g., via an intermediary computing unitsuch as a server, and/or via a storage device) with one or more of:

-   -   Client terminal(s) 218, for example, when computing unit 206        acts as a server providing SaaS.    -   A remotely located additive manufacturing system 220 that        manufactures the geometric object based on the generated set of        univariate curves. It is noted that 3D printer 202 may be        locally located and connected to computing unit 206 (e.g.,        integrated within, connected by a cable, connected by a short        range wireless communication protocol, connected by a local        network), and AM system 220 may be remotely located and        connected to computing unit 206 using network 214.    -   A storage unit 222 that stores the initial definition of the        geometric object (e.g., provided by client terminal 218, server        216, and/or used by computing unit 206 for remote storage)        and/or the generated set of univariate curves (e.g., created by        computing unit 206 for use, for example, by manufacturing system        220 for manufacturing the 3D object). Storage unit 222 may        include, for example, a storage server, a computing cloud        storage server, or other implementations.

Computing unit 206 includes or is in communication with a user interface224 allowing a user to enter data and/or view presented data. Exemplaryuser interfaces 224 include, for example, one or more of, a touchscreen,a display, a keyboard, a mouse, and voice activated software usingspeakers and microphone.

One or more acts of the method described with reference to FIG. 1A-B maybe implemented by code instructions 208A stored in program store 208(and/or stored in data repository 210) executed by processing unit 204of computing unit 206.

At 102, an initial definition of the 3D geometric object is received bycomputing unit 206. For example, the initial definition may be designedby a user using a design package using user interface 224, obtained fromlocal or remote storage, transmitted by remote client 218, and the like.

Optionally, the 3D geometric object is represented as a set of(optionally trimmed) surfaces boundary representation (B-rep).Alternatively or additionally, the 3D geometric object is represented asa set of piecewise linear polygonal meshes B-rep. Alternatively oradditionally, the 3D geometric object is represented as a trivariateparametric volume representation. Alternatively or additionally, the 3Dgeometric object is represented as a trimmed trivariate parametricvolume representation, for example, as described with reference to FadyMassarwi and Gershon Elber. A B-spline based framework for volumetricobject modeling. Computer Aided Design, Vol 78, pp 36-47, September2016.

At 108, the 3D geometric object is covered with at least one non-planarunivariate curve, to create a covering set of univariate curves within atolerance requirement.

The set of univariate curves are general 3D curves, which is notablydifferent than standard AM methods in which the print path are locatedalong parallel slices that are filled with lines.

One or more curves may be inserted between two curves to satisfy thetolerance requirement when a distance metric between the two curves doesnot satisfy the tolerance requirement.

In terms of formal mathematic representation, consider the volumeenclosed by a 3D geometric object, V. The desired tolerance (i.e.,tolerance requirement) is denoted as ε∈

. A valid curve covering of V is denoted by a set of n univariateparametric curves C={C₁(t) . . . C_(n)(t)}, so that for any pointp_(v)∈V there exists a point p_(c)∈C_(i)(t), C_(i)(t)∈C, for which∥p_(c)−p_(v)∥≤ε.

At 114, a representation of the 3D geometric object is generated basedon the set of univariate curves. The set of univariate curves fill thevolume of the 3D geometric object. The univariate curves may be used asprint paths by a printing head of a 3D printer to create a physicalrepresentation of the 3D geometric object.

Reference is now made to FIG. 8, which is a schematic depicting anexample of a helical object 802 (i.e., the received 3D geometric object)and a set of nine univariate curves 804 created using the systems and/ormethods described herein, in accordance with some embodiments of thepresent invention. It is noted that the curves of set 804 naturallyfollow the shape of (the center axis of) a helix. Using set 804 astool-paths to manufacturing the helix (e.g., using AM, or computernumerical control (CNC)) creates a physical object with improvedmechanical strength and/or improved surface finish, for example,compared to slicing based print-paths that are parallel to the XY planeperpendicular to the printing head).

It is noted that the curves of the 3D object are not necessarily limitedto the parameterization in which the 3D geometric object is given if atall defined. In cases where a more general set of covering curves isneeded, an external directional vector field that enclosed the 3D objectmay be defined to provide desired directions of the covering curves. Forexample and in addition, other external fields may prescribe attributesof the physically manufactured object, for example, color, transparency,desired materials' composition, mechanical strength and/or surfacefinish. The systems and/or methods described herein may operate usingthe directional vector field (that encompasses the entire geometry orpart thereof), to compute the set of curves for the 3D geometric objectthat conform to the prescribed external vector field (e.g., directions).The directional field may optionally be a (possibly trimmed) trivariate.

Reference is now made to FIG. 1B, which provides additional detailand/or additional features to the method discussed with reference toFIG. 1A.

At 102, the definition of the 3D geometric object is received, asdiscussed with reference to FIG. 1A.

Optionally, the received 3D geometric object may be a trimmed trivariatevolume. Trimmed trivariate volumes are defined similarly to trimmedsurfaces, for example, as a trivariate function with a domain that is asubset of the box domain of the tensor product volume, as described withreference to Elaine Cohen, Richard F Riesenfeld, and Gershon Elber.Geometric modeling with splines: an introduction. A K Peters, Ltd.,2001, or as described with reference in Fady Massarwi and Gershon Elber.A B-spline based framework for volumetric object modeling. ComputerAided Design, Vol 78, pp 36-47, September 2016.

Reference is now made to FIG. 9, which is a schematic of an exemplarytrimmed helical volume 902 that is created by intersecting helix 802 ofFIG. 8 with a sphere, in accordance with some embodiments of the presentinvention. The set of curves may be computed for volume 902 using thesystems and/or methods described herein.

Additional examples and details with respect to trimmed trivariates arefound, for example, with reference to Fady Massarwi and Gershon Elber. AB-spline based framework for volumetric object modeling. Computer AidedDesign, Vol 78, pp 36-47, September 2016.

Referring now back to block 102 of FIG. 1B, Optionally, an externaldirectional vector field is defined, for example, manually entered by auser, based on a predefined stored configuration, encoded as part of the3D geometric object, and/or automatically calculated by analysis code.The calculation of surfaces and/or calculation of univariate curves (asdescribed herein) is performed according to the external directionalvector field to create the set of univariate curves.

The external vector field, parameterization(s), and/or other defineddesign goals may be applied to the created set of curves, providing agenerality to the method described herein. As used herein, the termdesign streamlines, or streamlines, of the received geometric 3D object,denote the directions of the curves that cover the volume enclosed bythe 3D geometric object, V, based on a set of design goals. The coverageof the curves of the received 3D geometric object (as described herein)may be considered the design streamlines of the received 3D geometricobject with regards to the design goal(s). Exemplary representation ofthe external/additional vector field and/or the parameterization (e.g.,exemplary design goals) include: a surface finish, mechanical strength,color, transparency, material property and/or the directional field ofthe covering curves (the streamlines). The external vector field and/orparameterization create the set of curves that are streamlines of thereceived 3D object with respect to the design goals, without necessarilyrequirement modification to the covering method and/or system describedherein.

At 108, a set of univariate curves is computed for the 3D geometricobject, within the tolerance requirement. The univariate curves may becomputed using one or more of the exemplary methods described withreference to blocks 109A-109D.

Optionally, at block 109A, for a (possibly trimmed) trivariates' 3Dgeometric object, the covering set of univariate curves may be computedfor (possibly trimmed) trivariates using intermediate bivariateparametric (possibly trimmed) surfaces that cover the 3D geometric(possibly trimmed) trivariate, within the tolerance requirement. Each ofthe computed surfaces is then fitted with univariate curve(s).

In terms of formal mathematic representation, and when the 3D geometricobject is a (possibly trimmed) trivariate, the parametric trivariatevolume be denoted as V(u,v,w):D→

³, where D is a box domain in

³. The desired tolerance (i.e., tolerance requirement) is denoted as ε∈

. A valid curve covering of V is denoted by a set of n univariateparametric curves C={C₁(t) . . . C_(n)(t)}, so that for any pointp_(v)∈V(u,v,w) there exists a point p_(c)∈C_(i)(t), C_(i)(t)∈C, forwhich ∥p_(c)−p_(v)∥≤ε.

The valid surface coverage of the parametric trivariate volume V may bemathematically represented as a set of m bivariate parametric surfacesS={S₁(u,v) . . . S_(m)(u,v)}, so that for any point p_(v)∈V(u,v,w) thereexists a point p_(s)∈S_(i)(u,v), S_(i)(u,v)∈S, for which∥p_(s)−p_(v)∥≤ε.

An upper bound on the Hausdorff distance between two adjacent surfaces,that is based on an iso-distance notion may be defined. In terms ofmathematical representation, a vector field may be defined for twoparametric surfaces S₁(u,v) and S₂(u,v), sharing a common (u,v) domain,as D(u,v):=S₁(u,v)−S₂(u,v). The iso-distance between S₁ and S₂ may bedefined as Δ_(iso)(u,v):=∥D(u,v)∥. Since Δ_(iso)(u,v) is non-rational(due to the square root), Δ² _(iso)(u,v) (the iso-distance squared) isused instead. The Hausdorff distance between two surfaces S₁ and S₂ maybe represented as d_(H), and denoted by:

${d_{H}\left( {S_{1},S_{2}} \right)} = {\max \left\{ {{\sup\limits_{a \in S_{1}}\inf\limits_{b \in S_{2}}{{a - b}}},{\sup\limits_{p \in S_{2}}\inf\limits_{q \in S_{1}}{{p - q}}}} \right\}}$

A one sided point-surface Hausdorff distance may be denoted as:

${d_{Hs}\left( {{a \in S_{1}},S_{2}} \right)} = {\inf\limits_{b \in S_{2}}{{a - b}}}$

The iso-distance Δ_(iso)(u₀,v₀) for a pair of points denotes as a=S ¹(u₀,v₀), b=S₂(u₀,v₀) is an upper bound on the one sided point-surfaceHausdorff distance of both points d_(H)(a∈S₁, S₂), d_(H)(b∈S₂, S₁).

Computation of Δ² _(iso)(u,v) may be more efficient (e.g., in terms ofcomputation time, memory resources, and/or processing resources) thancomputation of the Hausdorff distances.

Reference is now made to FIG. 3, which is pseudo code of a method forcovering a, optionally trimmed, trivariate volume (i.e., V) usingiso-parametric surfaces of V, using Δ² _(iso)(u,v), in the w direction,in accordance with some embodiments of the present invention. Then, foreach iso-parametric surface, the iso-parametric curves are extractedfollowing the described method to cover a surface, as described withreference to Gershon Elber and Elaine Cohen. Adaptive isocurve-basedrendering for freeform surfaces. ACM Transactions on Graphics (TOG),15(3):249-263, 1996, extended to also handle trimmed surfaces.

When Δ² _(iso)(u,v) is computed (line 4), a determination is made ifanother surface is required between the two extreme surface to get avalid coverage. When Δ² _(iso)(u,v)<ε², ∀(u,v), no additional surfacesare added in between, and the method terminates. When Δ² _(iso)(u,v)>ε²for some (u,v) values, then a trimmed surface (line 12) is introduced.The trimmed surface is added between the two surfaces to satisfy thetolerance requirement when the distance metric between the two surfacesdoes not satisfy the tolerance requirement.

The trimmed surface is sized and positioned between the two surfacesaccording to where the distance metric between the two surfaces does notsatisfy the tolerance requirement, so that the insertion of the trimmedsurface does satisfy the tolerance requirement (or otherwise the processcontinues recursively). The tensor product surfaceS_(mid)(u,v)=V(u,v,w_(mid)) is trimmed to include (u,v) for which Δ²_(iso)(u,v)>ε².

When a middle surface is introduced, then the method is invokedrecursively for the newly created pair of adjacent surfaces (w_(low),w_(mid)) and (w_(mid), w_(high)) to further verity that the volumebetween them is covered.

When a valid coverage of V by surfaces is obtained (e.g., according tothe valid surface coverage described herein, and/or using the method ofFIG. 3), a curve coverage is created by covering each surface withcurves.

Each of the surfaces is fitted with one or more univariate curves tocreate a covering set of univariate curves to the surface.

The coverage of the surface using curves may be implemented based on themethod described with reference to FIG. 3, using curves in place of thesurfaces, starting with two surface boundary curves and recursivelyadding interior curves as to satisfy the tolerance requirement.

Reference is now made to FIG. 4, which is a schematic depicting theprocess of starting with a volume V (of a quarter of a torus with majorradius 1 and minor radius 0.2) 402, covering V with surfaces 404, andfitting curves 406 to the surfaces, in accordance with some embodimentsof the present invention. It is noted that schematic 404 depictsalternating full and trimmed surfaces, based on the method of insertingadditional trimmed surfaces between full surfaces described withreference to FIG. 3.

Reference is now made to FIG. 5, which is a schematic that depicts acoverage of a trivariate volume V(u,v,w) (of a quarter of a torus withmajor radius 1 and minor radius 0.2) covered by surfaces with differentvalues of the tolerance requirement (e), in accordance with someembodiments of the present invention. Schematic 502 includes surfacesfor e=0.4. Schematic 504 includes surfaces for e=0.1. Schematic 506includes surfaces for E=0.05.

The surface coverage is calculated using the method described withreference to FIG. 3. It is noted that m (of the method described withreference to FIG. 3) is defined to ensure a minimal depth of recursivecalls, for example, in case V(u,v,w) is a periodic trivariate (e.g., afull torus), where V(u,v,w_(min))=V(u,v,w_(max)). If Δ² _(iso)(u,v)<ε²,then any point on the line between V(u₀,v₀,w_(low)) andV(u₀,v₀,w_(high)) is covered as its distance to one of these surfacepoints is less than ε (in fact less than ε/2). However, Δ² _(iso)<ε² forany (u,v) does not imply that all points in the volume V(u,v,w_(r)),w_(r)ϵ(w_(low), w_(high)) are covered. If some point p_(w) on theparametric curve V(u₀,v₀,w_(r)) is sufficiently far from the linebetween V(u₀,v₀,w_(low)) and V(u₀,v₀,w_(high)) it is possible that p_(w)remains uncovered.

Reference is now made to FIG. 6, which is a schematic that depicts anexample of a situation in which S₁=V(u₀,v₀,w_(low)) andS₂=V(u₀,v₀,w_(high)) are close together, and satisfy tolerancerequirement ε (i.e., less than ε), but the volume in between thesurfaces includes points (along a w iso-parametric curve 602 between thesurfaces) that are too far from both surfaces, in accordance with someembodiments of the present invention.

Referring now back to FIG. 1, at 109A, an optional check is performedthat the set of computed curves and/or computed surfaces satisfy thetolerance requirement. The check includes verification that points onthe fitted curves (optionally all points on all curves) satisfy thetolerance requirement.

The check may be performed after computation of the respective curves,and/or simultaneously with computation of each curve, and/or at otherevents during the process.

An error of correctly determining the iso-distance measure ispotentially problematic especially on the boundaries. Different methodsare described herein to solve this technical problem of an invalidcoverage (e.g., for B-spline and Bezier volumes), by augmenting and/orreplacing the bound of Δ² _(iso) with the check of whether or notiso-surfaces V(u,v,w_(low)) and V(u,v,w_(high)) are a valid coverage ofthe convex hull of the control mesh of V(u,v,w_(r)), w_(r)∈(w_(low),w_(high)) that contains V (optionally represented as a Bezier orB-spline).

A valid coverage of the convex hull of the control mesh of the set ofunivariate curves that fill the volume of the 3D geometric objectbetween the two surfaces is provided.

In terms of mathematical representation, for CH(V) denoting the convexhull of the control mesh of (optionally a B-spline or a Bezier) volumeV(u,v,w), wϵ[w_(min), w_(max)]. When every point p_(h)ϵCH(V) is coveredby surfaces V(u,v,w_(min)) and V(u,v,w_(max)), then V(u,v,w) is coveredby V(u,v,w_(min)) and V(u,v,w_(max)).

Checking that the surfaces cover the convex hull may be performed, forexample, by dividing the trivariate volume (or portion thereof) into two(e.g., by splitting at w_(mid)) and checking that the convex hull of thecontrol mesh (of each of the split sub-volumes) is covered by thecorresponding surface. For non-singular volumes, the check may beperformed by checking the convex hull of the control meshes of the sixbounding surfaces (or five since one is the covering surface itself).

Other exemplary methods to ensure complete coverage may include bybounding the length of the univariate (e.g., iso-parametric) curves. Thebounding is according to the tolerance requirement between the twosurfaces corresponding to each curve. For example, given aniso-parametric curve from one covering surface to another, if the arclength of the curve is less than 2ε then full coverage is assured forall points on the line. Examining such bounds for all lines (throughtheir control poly-lines) may be used to ensure complete volumecoverage.

Optionally, a check is performed to identify over-coverage of a region.The over-coverage check may be performed by checking that correspondingpoints on the surfaces within the region are considered closer than theyactually are when the two surfaces are skewed relative to one anotherabove a skewing requirement. Checking for coverage of the convex hullmay provide a tighter bound on the Hausdorff distance than Δ²_(iso)(u,v).

Another exemplary method to prevent over-coverage is based on using adifferent measure of distance than the iso-distance (i.e., Δ²_(iso)(u,v) described with reference to FIG. 3). For example, in orderto account for the relative orientation of two surfaces (or curves), thefollowing may be defined:

${\Delta_{iso}^{new}\left( {u,v} \right)}:=\frac{{\left( {{D\left( {u,v} \right)},\left( {{n_{1}\left( {u,v} \right)} + {n_{2}\left( {u,v} \right)}} \right)} \right)}^{2}}{{\left( {{n_{1}\left( {u,v} \right)} + {n_{2}\left( {u,v} \right)}} \right)}^{2}}$

Where n₁(u,v) and n₂(u,v) are the unit normal fields of the surfacesaccounting for the relative skewing between the two surfaces. Using theabove (rational) measure, corresponding points on the two surfaces thatdiffer much in direction, with respect to the surface normals, areconsidered closer than they are, the logic being that other points (withother (u,v) values) on the other surface are closer.

Reference is now made to FIG. 7, which is a schematic of an examplewhere a new measure may work better than the Δ² _(iso) measure describedherein (which may be overly conservative), in accordance with someembodiments of the present invention. As shown in FIG. 7, for surfacesS₁ and S₂, the iso-distance Δ_(iso) may be arbitrarily larger than theone sided Hausdorff distance d_(Hs).

It is noted that the Δ_(iso) ^(new)(u,v) measure no longer bounds fromabove the Hausdorff distance, and may yield a non-valid coverage in somecases.

Referring now back to FIG. 1B, alternatively or additionally, at 109Bthe volume enclosed by a 3D object representation may be divided intomultiple discrete cells on a 3D grid, optionally using dexel (depthcells), with reference to Real-time shaded NC milling display, Tim VanHook, ACM SIGGRAPH Computer Graphics, Volume 20 Issue 4, August 1986,Pages 15-20.

Optionally, all accessible cells (those without any other cell belowthem) are uniquely covered by curves and removed from the 3D, a processthat continues until no more uncovered cells remain. Optionally, eachcell is covered by curve(s) by tracing a freeform curve from the center(or other location) of each respective discrete cell, optionallyfollowing a direction according to the external vector field. Thefreeform curve tracing terminates when another covered discrete cells isencountered, or when the freeform curve leaves the volume prescribed bythe respective discrete cell. The created set of curves covers the 3Dobject according to the tolerance requirement (ε) with boundedredundancy, and optionally following the external direction (i.e.,vector) field.

Alternatively or additionally, at 109C, when the external vector field(e.g., direction field) is defined, exemplary methods for computing thecurves to cover the trivariate volume may be implemented. Thedirectional field is optionally prescribed by an external trivariatevolume that encloses the given 3D geometric object. In such a case, thecovering curves follow the external trivariate volume when the coveringset of univariate curves is created, optionally spread uniformly oroptionally using the systems and/or methods described herein withreference to the trivariate parametric volume. The created set ofunivariate curves of the external field are clipped to fit inside thegiven 3D geometric object. Computation of the intersections of thecovering curves with the 3D geometric object, toward the clipping, maybe performed algebraically, by computing the intersection of coveringcurve C(t)=(C_(x)(t), C_(y)(t), C_(z)(t)) and the 3D geometric object'sboundary surface S(u,v)=(S_(x)(u,v), S_(y)(u,v), S_(z)(u,v)), andsolving for: C_(x) (t)=S_(x)(u,v), C_(y)(t)=S_(y)(u,v),C_(z)(t)=S_(z)(u,v), or using (e.g., piecewise linear, and/or piecewiseconstant) approximations of the boundary and the curve and computing theintersections using the approximations.

Reference is now made to FIG. 10, which is a schematic depicting theprocess of clipping the set of curves according to an external vectorfield, in accordance with some embodiments of the present invention.Image 1002 depicts the set of curves uniformly spread in an externalhelix trivariate directional field with a square cross sectional shapethat is to be clipped. Image 1004 is a zoom-in of the bottom of thesquare helix of image 1002. Image 1006 depicts the result of clippingthe set of curves for the square helix directional field to include thecurves that fit inside the helix with a circular cross section.

The external direction field may be used to achieve covering curves withcontrol over the uniformity of the distribution, or controlling otherproperties in the covering, including obviously the directions of thecurves.

Reference is now made to FIG. 19, which includes schematics depictingthe differences between the computed set of covering curves using theoriginal parameterization of a 3D trivariate object and the set ofcovering curves computed using an external vector field and clipping, inaccordance with some embodiments of the present invention. FIG. 19 showszoom-in portions of the end of the set of curves computed for a helix 3Dgeometric object using an external trivariate direction field (i.e.,external vector field) and/or created by clipping. Set of curves 1902are created using the original parameterization defined by the received3D trivariate object. Each tube represents the radius of the respectivecurve. Set of curves 1904 are created using an external direction fieldthat defines a square cross profile for the helix instead of the circleprofile. Set of curves 1906 are created by clipping set of curves 1904created using the external direction field to fit the original received3D geometric object.

Referring now back to FIG. 1B, alternatively or additionally, at 109Dthe set of curves may be created using function composition (e.g.,curve-trivariate function composition) to map the computed curves intoEuclidean space. The computed curves are thereby not necessarily limitedto the original parameterization. In terms of mathematicalrepresentation, C(t)=(u(t), v(t), w(t)) denotes a curve in the domain oftrivariate T(u,v,w). T(C)=T(u(t), v(t), w(t)) denotes a curve in theEuclidean space. The set of curves may be computed by placing curves inthe domain of T, with density that may compensate for T beingnon-isometric.

Reference is now made to FIG. 11, which includes schematics depictingthe process of mapping curves into Euclidean space, in accordance withsome embodiments of the present invention. Image 1102 depicts the domain(a cube) of the trivariate paved with univariate curves (as two sets ofconcentric helices). Image 1104 depicts the image of 1102 composed withthe trivariate to yield the set of curves in Euclidean space.

Referring now back to FIG. 1B, at 112, the set of univariate curves areordered for printing by a 3D printing head using material depositionalong each curve-member of the set of univariate curves. The ordering isselected such that during the 3D printing process curves that are notyet printed may be printed when such curves have no unprinted curvesbelow.

The curves are ordered for printing to provide a technical solutionrelated to the technical problem of 3D printing a set of 3-space curvesin general 3D positions. At any point during the 3D-printing process,some parts of the 3D model are already printed, while other parts arewaiting to be printed. The printing head of the 3D printer has a certaingeometry that occupies a corresponding physical space. When the printinghead is unable to print the unprinted portion of the 3D model, withoutthe printing head penetrating (i.e., cutting into) an already printedpart, then the printing process cannot succeed. Either an unprinted partwill never be printed, or an already printed part will be gouged intoand destroyed by the printing head.

The technical printing problem arises when printing curves general 3Dcurves, in comparison to traditional slicing methods, in which thegeometry of the printing head may be assumed to occupy the half-spaceabove a plane parallel to the XY plane of the curves currently beingprinted. The printing head is consistently at a certain offset (in the zaxis direction) above the currently printed part. Since parts areprinted in planes parallel to the XY plane, from bottom to top, nopenetration of the printed parts by (the geometry of) the printing headoccurs, in slicing based 3D printing. The ordering of the curves of theset of univariate curves, one complete slice after another, ensures thatthe print-paths followed by the printing head do not result in gougingof already printed parts.

Going back to the full 3D printing of general univariate curves proposedherein, in terms of mathematical representation, a curve C_(i) is belowcurve C_(j) and denoted C_(i)<C_(j), if the geometry of the printinghead overlaps with curve C_(j) when printing curve C_(i). Symmetrically,curve C_(i) is above C_(j) if C_(j) is below C_(i). At any point in timeduring the 3D printing process, only the curves that are not yet printedand have no unprinted curves below them may be printed.

Optionally, the order is defined according to a computed a directedgraph that stores the possible valid ordering(s) for printing thecurve-members of the set of univariate curves. Each curve-member isassigned to a vertex of the directed graph. A directed edge between twovertexes of the graph is defined when a first of the two vertexes isbelow a second of the two vertexes relative to the 3D printing head.When the directed graph is acyclic the directed graph represents one ormore valid 3D printable ordering(s) of the set of univariate curves. Itis noted that different valid orderings may be represented by a commongraph.

Optionally, the directed graph stores the ordering of printing the setof univariate curves, where each curve-member is assigned to a vertex ofthe directed graph, and a directed edge between two vertexes exists whena first of the two vertexes is below a second of the two vertexesrelative to the 3D printing head, in 5-axis additive manufacturing 3Dprinting, at least in some printing head orientations. When the directedgraph is acyclic the directed graph represents a 3D printable orderingof the set of univariate curves.

In terms of mathematical representation, the directed graph denoted asG_(a)(V,E) is created for a set of n curves C={C₁(t) . . . C_(n)(t)} by:assigning each curve C_(i)∈C to a vertex v_(i)∈V. A directed edgee_(ij)∈E, from v_(i) to v_(j), exists if C_(i)<C_(j). G_(a) denotes theaccessibility graph for C. A cycle in the graph G_(a) is sometimestermed herein an inaccessibility cycle, representing animpossible-to-print set of curves. C represents a non-gouging printableset of curves if and only if G_(a) is acyclic.

G_(a) defines the set of possible correct printing orders for a set ofcurves. When a path exists from v_(i) to v_(j) in G_(a), curve C_(i)must be 3D-printed before curve C_(j). An inaccessibility cycleindicates that no printing order is found. The 3D-printing process maybe realized by subdividing the curves in C into a new set of curvesC_(new), that includes an acyclic G_(a). It is noted that the divisionmay be selected to reduce the number of curves that are subdivided. Acontinuous printing path (i.e., from non-divided curves) may yield astronger part than one made up of separate printed elements (i.e.,divided curves), for example, as described with reference to Sung-HoonAhn, Michael Montero, Dan Odell, Shad Roundy, and Paul K Wright.Anisotropic material properties of fused deposition modeling abs. RapidPrototyping Journal, 8(4):248-257, 2002.

For computing G_(a) for C, the geometry of the printing head may beprovided.

Reference is now made to FIG. 13, which is a schematic representing thegeometry of the printing head used to compute the order of the set ofcurves, in accordance with some embodiments of the present invention.The geometry of the printing head may be modeled, for example, as adownward (axis parallel to the z axis) pointing cone 1302. Cone 1302 maybe used as a bounding cone for different types of printing heads. Theangle of the cone is denoted as θ. The z-offset of the tip of cone 1302above the currently printed part 1304 denoted as z_(offs), is set tobound the geometry of the actual extruder in use.

To compute when curve C_(i)<C_(j), a check may be performed as towhether the volume that encloses the sweep of the cone along curveC_(i), intersects C_(j). An intersection indicates that C_(i)<C_(j).

Reference is now made to FIG. 14, which is a schematic depicting anintersection between a volume 1402 defined by a sweep of a cone boundingthe printing head during printing of a lower curve 1404, and a portionof an upper curve 1406, in accordance with some embodiments of thepresent invention. The intersection indicates that curve 1404 is belowcurve 1406, and therefore curve 1404 must be printed before curve 1406.

Referring now back to block 112 of FIG. 1B, the check that C_(i)<C_(j)may be performed for all pairs of curves in C to construct G_(a). Interms of computational efficiency, checking each pair of curves isO(n²). The computational efficiency of computing G_(a) may be improvedby calculating a conservative approximation of G_(a), denoted herein asG _(a). The conservative approximation may be computed, for example,using spatial division acceleration structures, for example a BVH(bounding volume hierarchy) described with reference to Steven M Rubinand Turner Whitted. A 3-dimensional representation for fast rendering ofcomplex scenes. In ACM SIGGRAPH Computer Graphics, volume 14, pages110-116. ACM, 1980, or a z-buffer, or based on other method. Thecomputation of a may be performed relatively efficiently once a curve issubdivided into (sufficiently for the acceleration structure) spatiallycompact sub-curves. For example, a BVH query may be used to find (aconservative approximation of) all sub-curves that are above othersub-curves. The data of which sub-curves are above other sub-curves isused to compute G _(a).

When G _(a) (corresponding to G_(a)) is calculated, an order may bedefined based on the valid possible ordering combinations. When G_(a) isacyclic, any order that concurs with the order imposed by G _(a) may beselected. Additional subdivision of the curves is not necessary. Forexample, the ordering may be implemented by initially identifying theaccessible curves (i.e., that have nothing below them) and adding theidentified accessible curves to the printing list (and removing themfrom G_(a)). Optionally, the 3D printer dynamically prints curves fromthe list while new curves are added to the list as the new curves becomeaccessible based on the printed curves. When the initial set of curvesmay be represented as an acyclic G_(a) the printing process willeventually terminate successfully.

Alternatively, one or more cycles are identified in the directed graphthat represents an impossible-to-print subset of univariate curves. Thecycle may be removed by dividing one or more of the subset members ofunivariate curves of the cycle to remove the cycle from the directedgraph. The cyclical directed graph is transformed into an acyclic graphby removal of the cycle. Ordering of the curve-members and printingaccording to the order may proceed as described herein.

The subdivision of the curves may be performed, for example, using atop-down, and/or bottom-up approach to identify where to subdivide thecurves. The top-down approach is based on starting with the generatedset of curves (as described herein). One or more curves are selectivelydivided (optionally selected to reduce or minimize the total number ofsubdivisions performed) and rebuild G _(a) until no cycles remain.

The bottom-up approach is based on starting with the curves subdividedinto small sub-curves termed herein fragments. The directed graph iscomputed by dividing each curve-member of the set of univariate curvesinto fragments. Each fragment has a length less than a definedfragmentation tolerance, denoted as δ. The fragmentation tolerance maycorrespond to the physically smallest curve that the printing head iscapable of printing, optionally a minimum height of material depositedby the 3D printing head. The fragments are iteratively merged(optionally as pairs of fragments), and/or previously merged fragments(i.e., curves of two or more fragments) may be additionally merged,while maintaining an acyclic directed graph.

In terms of mathematical representation,

={C₁ ^(l), . . . , C_(k) ^(l)} denotes a cycle in directed graph G _(a).A_(l)(C_(i) ^(l)) denotes the accumulated arc-length of C_(i) ^(l) in

. An assumption is made that no cycles made entirely of curves shorterthan δ exist. If

is a cycle, ∀i A_(l)(C_(i) ^(l))>δ. It is noted that cycles of curvesshorter than δ pose a local inaccessibility difficulty that may beresolved, for example, by expanding the bounding cone of the printinghead by δ.

The initial set of fragments are assumed to be representable as aacyclic G _(a). Pairs of fragments (and/or sub-curves created with aprevious merger) are merged while maintaining the acyclic G _(a). Themerging may be performed selectively to obtain a minimal final number ofcurves. The minimal final number of curves produces a stronger and/orfiner physical product.

Either approach may be shown to be an NP-complete problem (via areduction from the monotone planar 3-SAT problem described withreference to Mark De Berg and Amirali Khosravi. Optimal binary spacepartitions for segments in the plane. International Journal ofComputational Geometry & Applications, 22(03):187-205, 2012. Heuristicmethods may be used to further improve the computational efficiency offinding solutions. For the bottom-up approach, the curves are subdividedinto small fragments based on the fragmentation tolerance (as describedherein) such that each sub-curve is shorter than a predetermined value(e.g., corresponding to a layer height in slicing based AM, and/or theshortest curve that the printer is capable of handling while havingcycles of lengths<δ).

A greedy bottom-up approach may be used to merge the fragments accordingto the following instructions (which are coded into processor executableinstructions):

(1) Step 1:

-   -   (a) Initialize a list of sub-curves L to an empty list.    -   (b) Create the fragments by subdividing the initial set of        covering curves (until no sub-curve is longer than a given value        defined based on the fragmentation tolerance δ), and mark all        fragments as unprocessed.    -   (c) Using the current set of unprocessed fragments, and for each        original print-path curve C, find a single group of adjacent        fragments from C with no unprocessed fragments below the members        of the group (locally checking no edge in the G _(a) is directed        at the members of the group), and merge the members of the group        into a sub-curve. A heuristic method may be implemented to        select the best sub-curve based on the sub-curve with the least        amount of endpoints adjacent to unprocessed fragments (i.e.,        zero, one, or two), and breaking any ties by optionally        selecting the longer (in the parametric domain) sub-curve.    -   (d) Mark all fragments in the selected sub-curve as processed.        Add the sub-curve to L.    -   (e) Go to 1(c) unless no unprocessed fragments remain. The        lowest (z value) unprocessed fragment consistently remains a        possible choice in 1(c) so the process is bound to eventually        terminate.

(2) Step 2:

-   -   (a) When possible, merge a single pair of adjacent sub-curves        (from the same original curve, C) in L, and update L. Merging is        possible when a global check shows the G _(a) graph for the        sub-curves resulting from the merge remains acyclic.    -   (b) Go to 2(a) unless no further merges are possible. Every        merge reduces the number of sub-curves by one, so the process is        bound to eventually terminate.

The above method produces an ordered list of (sub-)curves that may be 3Dprinted (e.g., using AM). Steps 1 and 2 merge together sub-curves tocreate longer sub-curves. Step 1 performs the merging in a local (andcomputationally more efficient) manner. Step 1 greedily decides thefollowing: assume unprocessed fragments are unprinted, and processedfragments are printed, what would be the best (according to theheuristic method described in step 1(c)) chain of fragments from thesame curve to print right now? The best chain is then merged into asub-curve and marked as processed. Step 1 is iterated until there are nomore unprocessed fragments, and fragments (all, or most) have beenmerged into longer sub-curves. Step 2 resolves a more global question:When merging two sub-curves from the same original curve C, would theresulting set of curves have an acyclic G _(a)? When the answer is yes,then merging is performed.

Reference is now made to FIG. 15, which is a schematic that graphicallydepicts the process of merging sub-curves (and/or fragments), inaccordance with some embodiments of the present invention. Based on theabove described method, the first time step 1(c) is executed, fragments(represented as lines 1502 between points 1504) are initially mergedinto sub-curve 1506 (represented by the dashed lines between points).During the second iteration of step 1(c), fragments are merged to createsub-curve 1508 (represented by the solid line between points). Duringthe third iteration of step 1(c), fragments are merged to createsub-curve 1510 (represented by the dotted line between points). Duringthe fourth iteration of step 1(c), fragments are merged to createsub-curve 1512 (represented by the grey solid line between points).

In the second step, sub-curves 1506 and 1510 are merged, since themerged curve (of sub-curves 1506 and 1510) may be printed withoutcreating an inaccessibility cycle. It is noted that step 2 may beperformed on the fragments themselves (rather than on the sub-curvescreated by merging fragments in the previous steps). However, thedescribed method based on a greedy heuristic provides a morecomputationally efficient process for merging of fragments.

Reference is now made to FIG. 16, which is a schematic depicting theprocess of ordering the curves for printing a helical object covered bya set of nine curves, in accordance with some embodiments of the presentinvention. The curves covering the helical object are printed based onthe process of ordering the curves (as described herein) that preventsgouging of lower curves as the helix is being printed. Image 1602depicts the printed first 9 segments. Image 1604 depicts the printedfirst 18 segments. Image 1606 depicts the printed first 27 segments.Image 1608 depicts the printed first 36 segments, completing the model.

Optionally, each curve-member of the set of univariate curves isassigned a value for a parameter (optionally an effective coverageradius) denoting the amount of local material for deposition by a 3Dprinting head when the respective curve-member is used as a printingpath. The parameter may be variably defined along the length of eachcurve-member, for example, the value may vary for each point along thelength. The assignment of the value provides a technical solution to thetechnical problem of controlling the amount of deposited material alongthe path. For example, in traditional AM methods, covering the samelocation with more than one covering curve is problematic: when a pointis covered more than once, material will also be deposited theremultiple times, resulting in excess material being placed. The value forthe parameter for each curve is selected by taking into account thatprint-paths based on the set of curves (created by the systems and/ormethods described herein) that are not necessarily parallel to eachother, and the distance between adjacent curves may vary along theirlengths. To reduce the amount of excess material extruded when locationsare covered by multiple (or too closely packed) curves, the volume ofeach curve is set in a way that ensures locations are not covered by toomany curves, and consequently applied too much material.

Optionally, each curve (at one or more points) is assigned a value of aparameter denoting an effective coverage radius indicating the localmaterial deposition radius. Optionally, when the deposited materialalong the print path is implemented by tracing a virtual tube, theeffective coverage radius sets the local radius of the tube (optionally,the cross section of the tube is not circular).

The radius for each curve may be computed, for example, based on theprocess of the subdivision into sub-curves, similar to the describeddivision processed used to approximate G_(a). For each sub-curve, theclosest sub-curves from other curves are found, and the radius is setaccording to those neighboring sub-curves. Optionally, the radius is setto be half the distance to the closest neighbor. However, in some cases,the half distance may be too conservative.

Reference is now made to FIG. 17, which is a schematic depicting theprocess of selecting the material deposition radius for curves, inaccordance with some embodiments of the present invention. FIG. 17 mayrepresent a close-up of the nine curves of the helical object of FIG.16. The outer ring of curves 1702 are closer to one another than to acentral curve 1704. An excessive amount of empty space may result aroundcentral curve 1704. For example, the excessive amount of empty space mayresult when the effective coverage radius of curves 1702 and 1704 is setto half the distance to the closest neighbor. The radius of centralcurve 1704 is set according to the distance between point p and point r,where r=(p+q)/2, where q denotes a point on the upper exterior curve1702. Empty space is left around central tube 1704 when the radiusvalues for exterior tubes 1702 are set according to the distance fromother exterior tubes 1702 rather than to central curve 1704.

The empty space may be reduced or prevented, for example, by selectingthe radius for curves 1702 and 1704 so that each curve touches another.However, given the underlying packing issue it may be difficult tocompletely fill the entire volume. Alternatively, the 2D space may beapproximated in planar normal sections along the sub-curve by computing(an approximation of) the corresponding 2D Voronoi cell, for example,and setting the radius of each curve according to the area of therespective Voronoi cell. Optionally, printing paths with non-circularsections, as is usually the case, should be considered.

Optionally, a support volume is defined for the set of univariatecurves. The support elements are designated to be printed beforeprinting a portion of the set of univariate curves located just abovethe respective support elements. The support elements help ensure thatthe printed parts of the 3D object are supported, and that the orderbetween the 3D object and the support elements is maintained. Thesupport volume may be computed, for example, using ray casting. Forexample, using vertical rays, in a grid, using, optionally a tessellatedversion of, the boundary surfaces of the received 3D object. The supportvolume is filled using support elements according to a defined density.For example, using L-shaped curves, a grid, or other implementations.Additional detail of minimal support elements used in AM may be found,for example, with reference to Ben Ezair, Fady Massarwi, and GershonElber. Orientation analysis of 3d objects toward minimal support volumein 3d-printing. Computers & Graphics, 51:117-124, 2015.

Reference is now made to FIG. 18, which is a schematic depictingL-shaped support elements, in accordance with some embodiments of thepresent invention. Schematic 1802 is a close up of helical object andsupport structure 1202 of FIG. 12. Block 1804 is a further zoom-in viewof helix 1202 and L-shaped support structure 1806.

Referring now back to FIG. 1B, optionally, at 113, material (e.g.,graded) information is added in association with one or more curves, forexample, the material used when manufacturing graded-material 3Dobjects, 3D printing each curve. The material information may beprovided, for example, manually entered by a user, retrieved from astorage device storing exemplary material information, and/orautomatically generated by analysis code.

At 114, the representation of the 3D object is generated based on theset of curves, as described with reference to FIG. 1A. For example, theset of curves may be stored in a format suitable for printing the 3Dobject by an AM device.

Optionally, at 116, a physical object based on the 3D geometric objectis manufactured, for example, printed by an additive manufacturingsystem, using a CNC based system, or other manufacturing systems, usingthe set of univariate curves as printing paths and/or tool guidingpaths.

When every point in the received 3D geometric object is covered withinthe tolerance requirement, every location within the 3D object is 3Dprinted.

The systems and/or methods described herein, in accordance with someembodiments of the present invention, may be used to distribute curvesin a volume. Alternatively or additionally, fiber placement and/orcomposite material manufacturing is performed by using the set ofunivariate curves as layout paths.

The systems and/or methods (e.g., code executable by processor(s))described herein that create the set of univariate curves provide extrafreedom using the general representation of the curves, which producessuperior AM print-paths, and creates improved 3D models.

Reference is now made to FIG. 12, which is a schematic of the process ofcreating the set of curves that are used to 3D print the geometricobject, in accordance with some embodiments of the present invention.Image 1202 depicts a helix with support structure for processing intothe set of curves. Image 1204 depicts nine curves that are computed forthe helix of image 1202. The value of the radius of the curves controlthe diameter of each tube. Image 1206 includes about 270 covering curveswith radius values controlling the diameter of the depicted tubes, aftersplits due to accessibility analysis. Printing the helical object ofimage 1206 using the curves (which follow the general shape of thehelix) as print-paths creates an improved 3D printed model, for example,with improved surface finish and/or superior mechanical strength, forexample, in comparison to printing using standard slicing methods, forexample, as described with reference to Prashant Kulkarni, Anne Marsan,and Debasish Dutta. A review of process planning techniques in layeredmanufacturing. Rapid Prototyping Journal, 6(1):18-35, 2000.

It is noted that alternatively, the set of covering curves may be usedfor example, for geometric modeling of anisotropic heterogeneousmaterials such as composite materials, and/or placement of fibers.Having a (trimmed) trivariate representation, optional volumetric fieldinformation, such as material properties and/or stiffness requirements,might be specified and propagated via the general univariate curvecovering, to the 3D printer, fully supporting the AM of graded-,heterogeneous, and anisotropic-materials.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

It is expected that during the life of a patent maturing from thisapplication many relevant additive manufacturing system will bedeveloped and the scope of the term additive manufacturing system isintended to include all such new technologies a priori.

As used herein the term “about” refers to ±10%.

The terms “comprises”, “comprising”, “includes”, “including”, “having”and their conjugates mean “including but not limited to”. This termencompasses the terms “consisting of” and “consisting essentially of”.

The phrase “consisting essentially of” means that the composition ormethod may include additional ingredients and/or steps, but only if theadditional ingredients and/or steps do not materially alter the basicand novel characteristics of the claimed composition or method.

As used herein, the singular form “a”, “an” and “the” include pluralreferences unless the context clearly dictates otherwise. For example,the term “a compound” or “at least one compound” may include a pluralityof compounds, including mixtures thereof.

The word “exemplary” is used herein to mean “serving as an example,instance or illustration”. Any embodiment described as “exemplary” isnot necessarily to be construed as preferred or advantageous over otherembodiments and/or to exclude the incorporation of features from otherembodiments.

The word “optionally” is used herein to mean “is provided in someembodiments and not provided in other embodiments”. Any particularembodiment of the invention may include a plurality of “optional”features unless such features conflict.

Throughout this application, various embodiments of this invention maybe presented in a range format. It should be understood that thedescription in range format is merely for convenience and brevity andshould not be construed as an inflexible limitation on the scope of theinvention. Accordingly, the description of a range should be consideredto have specifically disclosed all the possible subranges as well asindividual numerical values within that range. For example, descriptionof a range such as from 1 to 6 should be considered to have specificallydisclosed subranges such as from 1 to 3, from 1 to 4, from 1 to 5, from2 to 4, from 2 to 6, from 3 to 6 etc., as well as individual numberswithin that range, for example, 1, 2, 3, 4, 5, and 6. This appliesregardless of the breadth of the range.

Whenever a numerical range is indicated herein, it is meant to includeany cited numeral (fractional or integral) within the indicated range.The phrases “ranging/ranges between” a first indicate number and asecond indicate number and “ranging/ranges from” a first indicate number“to” a second indicate number are used herein interchangeably and aremeant to include the first and second indicated numbers and all thefractional and integral numerals therebetween.

It is appreciated that certain features of the invention, which are, forclarity, described in the context of separate embodiments, may also beprovided in combination in a single embodiment. Conversely, variousfeatures of the invention, which are, for brevity, described in thecontext of a single embodiment, may also be provided separately or inany suitable subcombination or as suitable in any other describedembodiment of the invention. Certain features described in the contextof various embodiments are not to be considered essential features ofthose embodiments, unless the embodiment is inoperative without thoseelements.

Although the invention has been described in conjunction with specificembodiments thereof, it is evident that many alternatives, modificationsand variations will be apparent to those skilled in the art.Accordingly, it is intended to embrace all such alternatives,modifications and variations that fall within the spirit and broad scopeof the appended claims.

All publications, patents and patent applications mentioned in thisspecification are herein incorporated in their entirety by referenceinto the specification, to the same extent as if each individualpublication, patent or patent application was specifically andindividually indicated to be incorporated herein by reference. Inaddition, citation or identification of any reference in thisapplication shall not be construed as an admission that such referenceis available as prior art to the present invention. To the extent thatsection headings are used, they should not be construed as necessarilylimiting.

1. A method of representing the volume of a three dimensional (3D)object for additive manufacturing of the 3D object, comprising:receiving an initial definition of a 3D object representation;calculating a covering set of univariate curves, the covering setcomprising a plurality of non-planar univariate curves, wherein thecovering set of univariate curves represent the volume of the 3D objectwithin a tolerance requirement; and generating a representation of the3D object based on the set of univariate curves, wherein the set ofunivariate curves represent the volume of the 3D object and defineprinting paths for a 3D printing head of an additive manufacturingsystem for additive manufacturing of the 3D object.
 2. The method ofclaim 1, wherein the covering set of univariate curves are calculatedbased on a plurality of non-planar bivariate parametric surfacescovering wherein the 3D object, wherein the 3D object comprises at leastone trivariate, within the tolerance requirement, and further comprisingfitting each of the plurality of surfaces with at least one univariatecurve.
 3. The method of claim 1, further comprising defining at leastone external directional vector field enclosing the 3D object, andwherein covering comprises fitting each of the plurality of curvesaccording to the at least one external directional vector field tocreate the covering set of univariate curves, and clipping the pluralityof curves to be inside the boundaries of the 3D object.
 4. (canceled) 5.The method of claim 1, wherein covering comprises dividing the volumeenclosed by the boundaries of the 3D representation into a plurality ofdiscrete volumetric cells, covering each cell by tracing a freeformcurve from the center of each respective discrete cell, wherein allaccessible cells without any other cell below them are uniquely coveredby curves and removed from the 3D object, until no more uncovered cellsremain, and further comprising terminating the tracing of the freeformcurve when another covered discrete cell is encountered or when thefreeform curve leaves the volume of the respective discrete cell.
 6. Themethod of claim 5, wherein the volume enclosed by the boundaries of the3D representation is divided into dexel (depth) cells.
 7. The method ofclaim 5, wherein the freeform curve is traced following a directionaccording to an external vector field.
 8. The method of claim 1, whereincovering comprises creating a set of univariate curves in the domain ofeach trivariate in the 3D object and mapping the curves, usingcurve-trivariate function composition, to Euclidean space.
 9. The methodof claim 1, wherein the 3D object comprises at least one member of thegroup consisting of: at least one trimmed trivariate, at least onetrivariate.
 10. (canceled)
 11. The method of claim 1, further comprisingperforming at least one of fiber placement and composite materialmanufacturing using the set of univariate curves as layout paths. 12.The method of claim 2, wherein the plurality of non-planar bivariateparametric surfaces are iso-parametric surfaces of a trivariateparametric volume representation of the 3D object representation,wherein the at least one univariate curve is an iso-parametric curve ofa trivariate parametric volume representation of the 3D objectrepresentation.
 13. (canceled)
 14. The method of claim 3, wherein the atleast one external directional vector field represents one or moreproperty selected from the group consisting of: color, transparency, asurface finish, a mechanical strength or stiffness, and graded-,anisotropic-, heterogeneous-material properties, and each of such fieldfurther comprising assigning to each curve-member of the set of coveringunivariate curves, a value for a parameter denoting the field's valuewhen the respective curve-member is used as a printing path, wherein theparameter is variable along the length of each curve-member.
 15. Themethod of claim 1, further comprising ordering the set of univariatecurves for printing by the 3D printing head along each curve-member ofthe set of univariate curves, wherein the ordering is selected such thatduring the 3D printing process curves that are not yet printed and haveno unprinted curves below them are printed.
 16. The method of claim 15,further comprising computing a directed graph that stores the orderingof printing the set of univariate curves, wherein each curve-member isassigned to a vertex of the directed graph, and a directed edge betweentwo vertexes exists when a first of the two vertexes is below a secondof the two vertexes relative to the 3D printing head, in full 3-axisadditive manufacturing 3D printing, wherein when the directed graph isacyclic the directed graph represents a 3D printable ordering of the setof univariate curves.
 17. The method of claim 15, further comprisingcomputing a directed graph that stores the ordering of printing the setof univariate curves, wherein each curve-member is assigned to a vertexof the directed graph, and a directed edge between two vertexes existswhen a first of the two vertexes is below a second of the two vertexesrelative to the 3D printing head, in 5-axis additive manufacturing 3Dprinting, wherein when the directed graph is acyclic the directed graphrepresents a 3D printable ordering of the set of univariate curves. 18.The method of claim 16, further comprising, identifying a cycle in thedirected graph that represents an impossible-to-print subset ofunivariate curves, and dividing at least one of the subset of univariatecurves to remove the cycle from the directed graph.
 19. The method ofclaim 16, wherein the directed graph is computed by dividing eachcurve-member of the set of univariate curves into a plurality offragments each having a length less than a fragmentation tolerancecorresponding to a minimum height of material deposited by the 3Dprinting head, and iteratively merging pairs of the plurality offragments while maintaining an acyclic directed graph.
 20. (canceled)21. The method of claim 1, further comprising assigning to eachcurve-member of the set of univariate curves, a value for a parameterdenoting an effective coverage radius indicative of the amount of localmaterial for deposition by the 3D printing head when the respectivecurve-member is used as a printing path, wherein the parameter isvariable along the length of each curve-member.
 22. The method of claim1, further defining a support volume for the set of covering univariatecurves, and representing the support volume using support elements,wherein the support elements are printed before printing a portion ofthe set of univariate curves.
 23. A system for processing of a threedimensional (3D) object for manufacturing, comprising: a program storestoring code; and a processor coupled to the program store forimplementing the stored code, the code comprising: code to receive aninitial definition of the 3D object, calculate a covering set ofunivariate curves, the covering set comprising a plurality of non-planarunivariate curves, wherein the covering set of univariate curvesrepresent the volume of the 3D object within a tolerance requirement;and generate a representation of the 3D object based on the covering setof univariate curves, wherein the covering set of univariate curvesrepresents the volume defined by the 3D object and define printing pathsfor a 3D printing head of an additive manufacturing system for additivemanufacturing of the 3D object.
 24. A computer program productcomprising a non-transitory computer readable storage medium storingprogram code thereon for implementation by a processor of a system forprocessing of a 3D object for additive manufacturing of the 3D object,comprising: instructions to receive an initial definition of the 3Dobject; instructions to calculate a covering set of univariate curves,the covering set comprising a plurality of non-planar univariate curves,wherein the covering set of univariate curves represent the volume ofthe 3D object within a tolerance requirement; and instructions togenerate a representation of the 3D object based on the covering set ofunivariate curves, wherein the covering set of univariate curvesrepresent the volume define by the 3D object and define printing pathsfor a 3D printing head of an additive manufacturing system for additivemanufacturing of the 3D object.